In this paper, stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs) is discussed. In contrast to the white noise examined in previous literature, we incorporate colored noise into the highly nonlinear hybrid NSDDEs. Under some assumptions, we can show that highly nonlinear hybrid NSDDEs have a unique global solution. Meanwhile, we establish some criteria related to noise-to-state stability (NSS) of global solutions. Additionally, some theorems are given to guarantee asymptotic stability in $L^{\hat{\alpha }}$ and almost surely asymptotic stability of global solution. These related discriminant rules are delay-dependent. Finally, an example is provided to demonstrate the validity of theoretical results.

This paper aims to formulate a class of nonlinear hybrid stochastic time-delay neural networks (STDNNs) with Lévy noise. Specifically, the coefficients of networks grow polynomially instead of linearly, and the time delay of given neural networks is non-differentiable. In many practical situations, nonlinear hybrid STDNNs with Lévy noise are unstable. Hence, this paper uses feedback control based on discrete-time state and mode observations to stabilize the considered nonlinear hybrid STDNNs with Lévy noise. Then, we establish stabilization criteria of $H_{\mathrm{\infty }}$ stability, asymptotic stability, and exponential stability for the controlled nonlinear hybrid STDNNs with Lévy noise. Finally, a numerical example illustrating the usefulness of theoretical results is provided.